Key Concepts and Formulas

• Cube Roots of Unity: The complex number $\omega = \frac{-1 + i\sqrt{3}}{2}$ is a non-real cube root of unity, satisfying $\omega^3 = 1$ and $1 + \omega + \omega^2 = 0$.
• Geometric Progression (GP) Sum: The sum of the first $n$ terms of a GP with first term $a$ and common ratio $r$ is given by $S_n = \frac{a(1 - r^n)}{1 - r}$, provided $r \neq 1$.
• Quadratic Equation from Roots: If $\alpha$ and $\beta$ are the roots of a quadratic equation, the equation can be written as $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.

Step-by-Step Solution

Step 1: Identify and Substitute the Cube Root of Unity

We are given $\alpha = \frac{-1 + i\sqrt{3}}{2}$. This is a standard representation of $\omega$, a non-real cube root of unity. We will substitute $\alpha = \omega$ throughout the problem to leverage the properties of cube roots of unity.

Step 2: Simplify the Expression for a

We are given $a = (1 + \alpha)\sum_{k=0}^{100} \alpha^{2k}$. Substituting $\alpha = \omega$, we get: $a = (1 + \omega)\sum_{k=0}^{100} \omega^{2k}$

Step 3: Simplify $(1 + \omega)$ using Cube Root of Unity Property

Using the property $1 + \omega + \omega^2 = 0$, we have $1 + \omega = -\omega^2$. Therefore, $a = (-\omega^2)\sum_{k=0}^{100} \omega^{2k}$

Step 4: Evaluate the Summation $\sum_{k=0}^{100} \omega^{2k}$

The summation represents a Geometric Progression (GP): $\sum_{k=0}^{100} \omega^{2k} = 1 + \omega^2 + \omega^4 + \omega^6 + \dots + \omega^{200}$ Here, the first term is $A = 1$, the common ratio is $R = \omega^2$, and the number of terms is $N = 101$. Using the formula for the sum of a GP: $S_{101} = \frac{1(1 - (\omega^2)^{101})}{1 - \omega^2} = \frac{1 - \omega^{202}}{1 - \omega^2}$

Step 5: Simplify $\omega^{202}$ using Cube Root of Unity Property

Since $\omega^3 = 1$, we can simplify $\omega^{202}$. Dividing 202 by 3, we get $202 = 3 \cdot 67 + 1$. Therefore, $\omega^{202} = \omega^{3 \cdot 67 + 1} = (\omega^3)^{67} \cdot \omega^1 = 1^{67} \cdot \omega = \omega$ Substituting back into the sum: $\sum_{k=0}^{100} \omega^{2k} = \frac{1 - \omega}{1 - \omega^2}$

Step 6: Further Simplify the Summation

We can factor the denominator using the difference of squares: $1 - \omega^2 = (1 - \omega)(1 + \omega)$. So, $\sum_{k=0}^{100} \omega^{2k} = \frac{1 - \omega}{(1 - \omega)(1 + \omega)}$ Since $\omega \neq 1$, we can cancel the $(1 - \omega)$ terms: $\sum_{k=0}^{100} \omega^{2k} = \frac{1}{1 + \omega}$

Step 7: Substitute the Simplified Summation Back into the Expression for a

Recall $a = (-\omega^2) \sum_{k=0}^{100} \omega^{2k}$. Therefore, $a = (-\omega^2) \cdot \frac{1}{1 + \omega}$

Step 8: Use Cube Root of Unity Property Again

Since $1 + \omega = -\omega^2$, we have: $a = (-\omega^2) \cdot \frac{1}{-\omega^2} = 1$ Thus, $a = 1$.

Step 9: Simplify the Expression for b

We are given $b = \sum_{k=0}^{100} \alpha^{3k}$. Substituting $\alpha = \omega$, we get: $b = \sum_{k=0}^{100} \omega^{3k}$

Step 10: Simplify $\omega^{3k}$ using Cube Root of Unity Property

Since $\omega^3 = 1$, we have $\omega^{3k} = (\omega^3)^k = 1^k = 1$. Therefore, $b = \sum_{k=0}^{100} 1$

Step 11: Evaluate the Summation for b

The summation is simply the sum of 1, 101 times (from $k=0$ to $k=100$). $b = \sum_{k=0}^{100} 1 = 101$ Thus, $b = 101$.

Step 12: Form the Quadratic Equation

We have found the roots $a = 1$ and $b = 101$. The sum of the roots is $a + b = 1 + 101 = 102$, and the product of the roots is $a \cdot b = 1 \cdot 101 = 101$. Therefore, the quadratic equation is: $x^2 - (a + b)x + ab = 0$ $x^2 - 102x + 101 = 0$

Common Mistakes & Tips

• Incorrect GP sum: Make sure to use the correct formula for the sum of a GP, especially paying attention to the number of terms and the common ratio.
• Assuming GP sum is always zero: The sum of a GP involving cube roots of unity is only zero if the number of terms is a multiple of 3 and the common ratio cycles through $1, \omega, \omega^2$.
• Forgetting the $k=0$ term: When summing from $k=0$, remember that the first term corresponds to $k=0$, which can be easily overlooked.

Summary

By recognizing $\alpha$ as the cube root of unity $\omega$, and utilizing the properties of $\omega$ and the GP sum formula, we simplified the expressions for $a$ and $b$ to find $a=1$ and $b=101$. Then, using the formula for a quadratic equation given its roots, we found the equation to be $x^2 - 102x + 101 = 0$.

Final Answer

The final answer is \boxed{x^2 - 102x + 101 = 0}, which corresponds to option (C).